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Theorem tfrcldm 6226
Description: Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.)
Hypotheses
Ref Expression
tfrcl.f  |-  F  = recs ( G )
tfrcl.g  |-  ( ph  ->  Fun  G )
tfrcl.x  |-  ( ph  ->  Ord  X )
tfrcl.ex  |-  ( (
ph  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
tfrcl.u  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
tfrcl.yx  |-  ( ph  ->  Y  e.  U. X
)
Assertion
Ref Expression
tfrcldm  |-  ( ph  ->  Y  e.  dom  F
)
Distinct variable groups:    f, G, x    S, f, x    f, X, x    f, Y, x    ph, f, x
Allowed substitution hints:    F( x, f)

Proof of Theorem tfrcldm
Dummy variables  z  a  b  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrcl.yx . . 3  |-  ( ph  ->  Y  e.  U. X
)
2 eluni 3707 . . 3  |-  ( Y  e.  U. X  <->  E. z
( Y  e.  z  /\  z  e.  X
) )
31, 2sylib 121 . 2  |-  ( ph  ->  E. z ( Y  e.  z  /\  z  e.  X ) )
4 tfrcl.f . . . 4  |-  F  = recs ( G )
5 tfrcl.g . . . . 5  |-  ( ph  ->  Fun  G )
65adantr 272 . . . 4  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Fun  G )
7 tfrcl.x . . . . 5  |-  ( ph  ->  Ord  X )
87adantr 272 . . . 4  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Ord  X )
9 tfrcl.ex . . . . 5  |-  ( (
ph  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
1093adant1r 1192 . . . 4  |-  ( ( ( ph  /\  ( Y  e.  z  /\  z  e.  X )
)  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
11 feq2 5224 . . . . . . . 8  |-  ( a  =  x  ->  (
f : a --> S  <-> 
f : x --> S ) )
12 raleq 2601 . . . . . . . 8  |-  ( a  =  x  ->  ( A. b  e.  a 
( f `  b
)  =  ( G `
 ( f  |`  b ) )  <->  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b ) ) ) )
1311, 12anbi12d 462 . . . . . . 7  |-  ( a  =  x  ->  (
( f : a --> S  /\  A. b  e.  a  ( f `  b )  =  ( G `  ( f  |`  b ) ) )  <-> 
( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b ) ) ) ) )
1413cbvrexv 2630 . . . . . 6  |-  ( E. a  e.  X  ( f : a --> S  /\  A. b  e.  a  ( f `  b )  =  ( G `  ( f  |`  b ) ) )  <->  E. x  e.  X  ( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b ) ) ) )
15 fveq2 5387 . . . . . . . . . 10  |-  ( b  =  y  ->  (
f `  b )  =  ( f `  y ) )
16 reseq2 4782 . . . . . . . . . . 11  |-  ( b  =  y  ->  (
f  |`  b )  =  ( f  |`  y
) )
1716fveq2d 5391 . . . . . . . . . 10  |-  ( b  =  y  ->  ( G `  ( f  |`  b ) )  =  ( G `  (
f  |`  y ) ) )
1815, 17eqeq12d 2130 . . . . . . . . 9  |-  ( b  =  y  ->  (
( f `  b
)  =  ( G `
 ( f  |`  b ) )  <->  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) )
1918cbvralv 2629 . . . . . . . 8  |-  ( A. b  e.  x  (
f `  b )  =  ( G `  ( f  |`  b
) )  <->  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) )
2019anbi2i 450 . . . . . . 7  |-  ( ( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b
) ) )  <->  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) )
2120rexbii 2417 . . . . . 6  |-  ( E. x  e.  X  ( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b
) ) )  <->  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) )
2214, 21bitri 183 . . . . 5  |-  ( E. a  e.  X  ( f : a --> S  /\  A. b  e.  a  ( f `  b )  =  ( G `  ( f  |`  b ) ) )  <->  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) )
2322abbii 2231 . . . 4  |-  { f  |  E. a  e.  X  ( f : a --> S  /\  A. b  e.  a  (
f `  b )  =  ( G `  ( f  |`  b
) ) ) }  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
24 tfrcl.u . . . . 5  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
2524adantlr 466 . . . 4  |-  ( ( ( ph  /\  ( Y  e.  z  /\  z  e.  X )
)  /\  x  e.  U. X )  ->  suc  x  e.  X )
26 simprr 504 . . . 4  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  -> 
z  e.  X )
274, 6, 8, 10, 23, 25, 26tfrcllemres 6225 . . 3  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  -> 
z  C_  dom  F )
28 simprl 503 . . 3  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Y  e.  z )
2927, 28sseldd 3066 . 2  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Y  e.  dom  F )
303, 29exlimddv 1852 1  |-  ( ph  ->  Y  e.  dom  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   A.wral 2391   E.wrex 2392   U.cuni 3704   Ord word 4252   suc csuc 4255   dom cdm 4507    |` cres 4509   Fun wfun 5085   -->wf 5087   ` cfv 5091  recscrecs 6167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfrcl  6227  frecfcllem  6267  frecsuclem  6269
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